Skip to main content Accessibility help
×
×
Home

Admissibility for a Class of Quasiregular Representations

  • Bradley N. Currey (a1)
Abstract

Given a semidirect product G = NH where N is nilpotent, connected, simply connected and normal in G and where H is a vector group for which ad() is completely reducible and R-split, let τ denote the quasiregular representation of G in L 2(N). An element ψL 2(N) is said to be admissible if the wavelet transform f ⟼ 〈 f, τ(·)ψ 〉 defines an isometry from L 2(N) into L 2(G). In this paper we give an explicit construction of admissible vectors in the case where G is not unimodular and the stabilizers in H of its action on are almost everywhere trivial. In this situation we prove orthogonality relations and we construct an explicit decomposition of L 2(G) into G-invariant, multiplicity-free subspaces each of which is the image of a wavelet transform . We also show that, with the assumption of (almost-everywhere) trivial stabilizers, non-unimodularity is necessary for the existence of admissible vectors.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Admissibility for a Class of Quasiregular Representations
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Admissibility for a Class of Quasiregular Representations
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Admissibility for a Class of Quasiregular Representations
      Available formats
      ×
Copyright
References
Hide All
[1] Arnal, D.,BenAmmar, M., Currey, B., and Dali, B., Explicit construction of canonical coordinates for completely solvable Lie groups. J. Lie Theory 15(2005), 521560.
[2] Currey, B., A continuous trace composition sequence for C*(G) where G is an exponential solvable Lie group . Math. Nachr. 159(1992), 189212.
[3] Currey, B., An explicit Plancherel formula for completely solvable Lie groups. MichiganMath. J. 38(1991), no. 1, 7587.
[4] Currey, B., Explicit orbital parameters and the Plancherel measure for exponential Lie groups. Pacific J. Math. (to appear).
[5] Currey, B., Smooth decomposition of finite multiplicity monomial representations for a class of completely solvable homogeneous spaces. Pacific J. Math. 170(1995), no. 2, 429460.
[6] Currey, B., The structure of the space of coadjoint orbits of an exponential solvable Lie group . Trans. Amer. Math. Soc. 332(1992), no. 1, 241269.
[7] Currey, B. and Penney, R., The structure of the space of co-adjoint orbits of a completely solvable Lie group. Mich. Math. J. 36(1989), no. 2, 309320.
[8] Duflo, M. and Moore, C., On the regular representation of a nonunimodular locally compact group. J. Functional Analysis 21(1976), no. 2, 209243.
[9] Duflo, M. and Raïs, M., Sur l’analyse harmonique sur les groupes de Lie résolubles. Ann. Sci. École Norm. Sup. 9(1976), no. 1, 107144.
[10] Fabec, R., and Ólafsson, G., The continuous wavelet transform and symmetric spaces. Acta Applicandae Math. (to appear)
[11] Führ, H., Admissible vectors for the regular representation. Proc. Amer. Math. Soc. 130(2002), no. 10, 29592970.(electronic).
[12] Ishi, H., Wavelet transforms for semidirect product groups. J. Fourier Anal. Appl. 12(2006), no. 1, 3752.
[13] Laugesen, R. S., Weaver, N., Weiss, G., and Wilson, E. N., A characterization of the higher dimensional groups associated with continuous wavelets. J. Geom. Anal. 12(2002), no. 1, 89102.
[14] Lipsman, R., Harmonic analysis on exponential solvable homogeneous spaces: the algebraic or symmetric cases. Pacific. J. Math. 140(1989), no. 1, 117147.
[15] Lipsman, R., Induced representations of completely solvable Lie groups. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17(1990), no. 1, 127164.
[16] Lipsman, R., The multiplicity function on exponential and completely solvable homogeneous spaces. Geom. Dedicata 39(1991), no. 2, 155161.
[17] Lipsman, R. and Wolf, J., Canonical semi-invariants and the Plancherel formula for parabolic groups. Trans. Amer. Math. Soc. 269(1982), no. 1, 111131.
[18] Liu, H. and Peng, L., Admissible wavelets associated with the Heisenberg group. Pacific J. Math. 180(1997), no. 1, 101123.
[19] Pedersen, N. V., On the infinitesimal kernel of irreducible representations of nilpotent Lie groups. Bull. Soc. Math. France 112(1984), no. 4, 423467.
[20] Pukánszky, L., On the characters and the Plancherel formula of nilpotent Lie groups. J. Functional Analysis 1(1967), 255280.
[21] Pukánszky, L., On the unitary representations of exponential Lie groups. J. Functional Analysis 2(1968), 73113.
[22] Weiss, G. and Wilson, E. N., The mathematial theory of wavelets. In: Twentieth Century Harmonic Analysis – A Celebration, NATO Sci. Ser. II Math. Phys. Chem. 33, Kluwer Acad. Publ., Dordrecht, 2001, pp. 329366.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed